Second-Order Reversible Reactions

Size comparison for first- and second-order and reversible reactions... 

Any combination of first-order reactions can be simulated by extension of this procedure. Reversible reactions add only the feature that reacted species can be regenerated from their products. Second-order reactions introduce a new factor, for now two molecules must each be independently selected in order that reaction occur in the real situation the two molecules are in independent motion, and their collision must take place to cause reaction. We load the appropriate numbers of molecules into each of two grids. Now randomly select from the first grid, and then, separately, randomly select from the second grid. If in both selections a molecule exists at the respective selected sites, then reaction occurs and both are crossed out if only one of the two selections results in selection of a molecule, no reaction occurs. (Of course, if pseudo-first-order conditions apply, a second-order reaction can be handled just as is a first-order reaction.)... 

Suppose that experiments show that both the forward reaction and the reverse reaction are elementary second-order reactions. Then we would write the following rate laws ... 

Apply the bounding theorem to the reversible, second-order reaction... 

From a mathematical standpoint the various second-order reversible reactions are quite similar, so we will consider only the most general case—a mixed second-order reaction in which the initial system contains both reactant and product species. 

Micellar rate enhancements of bimolecular, non-solvolytic reactions are due largely to increased reactant concentrations at the micellar surface, and micelles should favor third- over second-order reactions. The benzidine rearrangement typically proceeds through a two-proton transition state (Shine, 1967 Banthorpe, 1979). The first step is a reversible pre-equilibrium and in the second step proton transfer may be concerted with N—N bond breaking (17) (Bunton and Rubin, 1976 Shine et al., 1982). Electron-donating substituents permit incursion of a one-proton mechanism, probably involving a pre-equilibrium step. 

A rate equation is required for this reaction taking place in dilute solution. It is expected that reaction will be pseudo first-order in the forward direction and second-order in reverse. The reaction is studied in a laboratory batch reactor starting with a solution of methyl acetate and with no products present. In one test, the initial concentration of methyl acetate was 0.05 kmol/m3 and the fraction hydrolysed at various times subsequently was ... 

The decomposition of hydrogen iodide is a reversible second order reaction in both directions ... 

P4.11.01. OPTIMUM TEMPERATURE PROFILE OF REVERSIBLE FIRST AND SECOND ORDER REACTIONS. 

For a second order reaction in a slab or a sphere, analytical solutions proceed in terms of elliptic functions beyond the solution of P7.03.ll, although a numerical solution throughout may be preferable. Such a numerical procedure is adopted in P7.03.19 for a second order reversible reaction. 

Despite the problems that can afflict experimental cyclic voltammograms, when the method for deriving standard redox potentials is used with caution it affords data that may be accurate within a few tens of mV (10 mV corresponds to about 1 kJ mol-1), as remarked by Tilset [335]. Kinetic shifts are usually the most important error source The deviation (A If) of the experimental peak potential from the reversible value can be quite large. However, it is possible to estimate AEp if the rate constant of the chemical reaction is available. For instance, in the case of a second order reaction (e.g., a radical dimerization) with a rate constant k, the value of AEV at 298.15 K is given by equation 16.24 [328,339] ... 

As for first-order following chemical reactions, if the dimerization reaction (that is a second-order reaction) is slow (i.e. k2 is small), or if the scan rate is very high, only the reversible electron transfer is effectively active. 

Most electrode reactions of interest to the organic electrochemist involve chemical reaction steps. These are often assumed to occur in a homogeneous solution, that is, not at the electrode surface itself. They are described by the usual chemical kinetic equations, for example, first- or second-order reactions and may be reversible (chemical reversibility) or irreversible. 

The schemes considered are only a few of the variety of combinations of consecutive first-order and second-order reactions possible including reversible and irreversible steps. Exact integrated rate expressions for systems of linked equilibria may be solved with computer programs. Examples other than those we have considered are rarely encountered however except in specific areas such as oscillating reactions or enzyme chemistry, and such complexity is to be avoided if at all possible. 

The second-order reaction with adsorption of the ligand (2.210) signifies the most complex cathodic stripping mechanism, which combines the voltammetric features of the reactions (2.205) and (2.208) [137]. For the electrochemically reversible case, the effect of the ligand concentration and its adsorption strength is identical as for reaction (2.205) and (2.208), respectively. A representative theoretical voltammo-gram of a quasireversible electrode reaction is shown in Fig. 2.86d. The dimensionless response is controlled by the electrode kinetic parameter m, the adsorption... 

A kinetic study of nitrous acid-catalyzed nitration of naphthalene with an excess of nitric acid in aqueous mixture of sulfuric and acetic acids (Leis et al. 1988) shows a transition from first-order to second-order kinetics with respect to naphthalene. (At this acidity, the rate of reaction through the nitronium ion is too slow to be significant the amount of nitrous acid is sufficient to make one-electron oxidation of naphthalene as the main reaction path.) The reaction that initially had the first-order in respect to naphthalene becomes the second-order reaction. The electron transfer from naphthalene to NO+ has an equilibrium (reversible) character. In excess of the substrate, the equilibrium shifts to the right. A cause of the shift is the stabilization of cation-radical by uncharged naphthalene. The stabilized cation-radical dimer (NaphH)2 is just involved in nitration ... 

Second-Order Reversible Reactions. For the bimolecular-type second-order reactions ... 

In 1959, the coordinated mercaptide ion in the gold(III) complex (4) was found to undergo rapid alkylation with methyl iodide and ethyl bromide (e.g. equation 3).9 The reaction has since been used to great effect particularly in nickel(II) (3-mercaptoamine complexes.10,11 It has been demonstrated by kinetic studies that alkylation occurs without dissociation of the sulfur atom from nickel. The binuclear nickel complex (5) underwent stepwise alkylation with methyl iodide, benzyl bromide and substituted benzyl chlorides in second order reactions (equation 4). Bridging sulfur atoms were unreactive, as would be expected. Relative rate data were consistent with SN2 attack of sulfur at the saturated carbon atoms of the alkyl halide. The mononuclear complex (6) yielded octahedral complexes on alkylation (equation 5), but the reaction was complicated by the independent reversible formation of the trinuclear complex (7). Further reactions of this type have been used to form new chelate rings (see Section 7.4.3.1). 

This discussion applies to an irreversible second-order reaction. For reversible reactions the relationships are more complex and are discussed in the texts by Sherwood et al. (1975) and by Danckwerts (1970). 

FIGURE 5.19 Degradation of 0.1 mg/L 5-aminolevulinic acid (ALA) followed by a reversible second-order reaction at pH 7.4 (200 mM NaHP04). [Graph reconstructed from data by Bunke et al., J. Pharrn. ScL, 89, 1335 (2000).]... 

The kinetics of the C step are not always first order or pseudo-first order. A second-order reaction will produce qualitatively similar effects to those described above. However, the relative magnitude of the reverse peak current associated with the E step and hence the extent of reversibility and the shift in peak potential will depend on the concentration of the electroactive species for an EC2 mechanism. A process of this type will have a reversible E step at low concentrations or fast scan rates and an irreversible E step at high concentrations or slow scan rates. An example of an EQ-type reaction (Bond et al., 1983, 1989) is the electrochemical oxidation of cobalt (III) tris(dithiocarbamates) (Co(S2CNR2)3) at platinum electrodes in dichloromethane/0.1 M (C4H9)4NPp6 [equations (44) and (45)]. 

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